Embedded model Embedded model in $\mathbb{P}^{4}$
$ 0 $ | $=$ | $ x z^{2} + y z^{2} + y z w $ |
| $=$ | $x z w + y z w + y w^{2}$ |
| $=$ | $x y z + y^{2} z + y^{2} w$ |
| $=$ | $x z t + y z t + y w t$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} y - 4 x^{4} z - 8 x^{2} y z^{2} + 4 x^{2} z^{3} - 10 y^{2} z^{3} + 11 y z^{4} - 3 z^{5} $ |
Weierstrass model Weierstrass model
$ y^{2} + x^{4} y $ | $=$ | $ -4x^{6} - 2x^{4} - 16x^{2} + 4 $ |
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Embedded model |
$(0:1:0:0:0)$, $(0:0:1:-1:1)$, $(0:0:0:0:1)$, $(0:0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^3}\cdot\frac{512xw^{10}-25600xw^{9}t-526080xw^{8}t^{2}-2606140xw^{7}t^{3}-3274700xw^{6}t^{4}+2133118xw^{5}t^{5}+4458910xw^{4}t^{6}+1298370xw^{3}t^{7}-34490xw^{2}t^{8}+58500xwt^{9}+32xt^{10}+200y^{9}t^{2}+2300y^{7}t^{4}+11250y^{5}t^{6}+28475y^{3}t^{8}-17408yzw^{9}-498688yzw^{8}t-3469120yzw^{7}t^{2}-9039512yzw^{6}t^{3}-8508392yzw^{5}t^{4}-3872112yzw^{4}t^{5}-1857040yzw^{3}t^{6}-975468yzw^{2}t^{7}-117428yzwt^{8}-25600yzt^{9}-34304yw^{10}-670208yw^{9}t-3395912yw^{8}t^{2}-4267084yw^{7}t^{3}+5656708yw^{6}t^{4}+13342846yw^{5}t^{5}+8014562yw^{4}t^{6}+1658034yw^{3}t^{7}+230772yw^{2}t^{8}-22822ywt^{9}+11732yt^{10}}{w^{2}(500xw^{7}t+1901xw^{6}t^{2}+2696xw^{5}t^{3}+1818xw^{4}t^{4}+560xw^{3}t^{5}+28xw^{2}t^{6}-24xwt^{7}-4xt^{8}+300yzw^{7}+1400yzw^{6}t+2366yzw^{5}t^{2}+1962yzw^{4}t^{3}+864yzw^{3}t^{4}+192yzw^{2}t^{5}+16yzwt^{6}+600yw^{8}+2600yw^{7}t+4233yw^{6}t^{2}+3309yw^{5}t^{3}+1174yw^{4}t^{4}+10yw^{3}t^{5}-132yw^{2}t^{6}-40ywt^{7}-4yt^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
40.72.3.b.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Equation of the image curve:
$0$ |
$=$ |
$ 4X^{4}Y-4X^{4}Z-8X^{2}YZ^{2}+4X^{2}Z^{3}-10Y^{2}Z^{3}+11YZ^{4}-3Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
40.72.3.b.1
:
$\displaystyle X$ |
$=$ |
$\displaystyle -y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle -4y^{2}z^{2}+22z^{4}-20z^{3}t$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.